Proj construction ( $Proj S$ )

by taking the degree-zero part of its localization on each standard affine open of Proj. Localization identifies these module sheaves on overlaps. The construction is exact, by exactness of localization and exactness of taking a fixed graded component.

Tensor compatibility of graded sheafification on degree-one-generated Proj ( $M \otimes_{O_{X}} N ≅ M \otimes_{S} N$ )

 0  0

Under the degree-one generation condition for Proj, cover

X = Proj S

by degree-one charts. The degree-one localization of a graded module identifies the two local modules in the displayed formula, using localization commutes with tensor products. The maps are natural and agree on overlaps, so they glue to an isomorphism of sheaves of modules. The analogous assertion can fail for general positively graded rings.

Twisting sheaf on Proj ( $O_{Proj S} (n) = S (n)$ )

 0  0

With the graded shift convention

S (n)_{d} = S_{n + d}

, this is the sheaf associated with a graded module

S (n)

. Under the degree-one generation condition for Proj, it is an invertible sheaf: on

D_{+} (f)

with

de g f = 1

, multiplication by

f^{n}

freely generates its local module for every integer

n

. Without that hypothesis, it need not be invertible. The twisting sheaf on projective space is the standard special case. Stacks Project, Section 27.10 records the hypotheses and local trivializations.

Irrelevant ideal of a graded ring ( $S_{+}$ )

 0  0

For a nonnegatively graded ring

S = ⨁_{d \geq 0} S_{d}

, the irrelevant ideal is

S_{+} = ⨁_{d > 0} S_{d} .

(1)

The points of

Proj S

are precisely the homogeneous prime ideals that do not contain

S_{+}

Standard affine open of Proj ( $D_{+} (f)$ )

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For a positive-degree homogeneous element

f \in S

, the standard open

D_{+} (f) = {p \in Proj S : f \in / p}

(1)

is the affine scheme

Spec (S_{f})_{0}

Morphism on Proj induced by a graded ring homomorphism

 0  0

A graded homomorphism

S \to T

induces a morphism to

Proj S

on the open subset of

Proj T

where the inverse image of a prime does not contain

S_{+}

. On

D_{+} (φ (f))

it is induced by

(S_{f})_{0} \to (T_{φ (f)})_{0}

Invariance of Proj under an eventual graded isomorphism

 0  0

If a graded homomorphism

S \to T

is an isomorphism in every sufficiently large degree, then it induces an isomorphism

Proj T \sim Proj S .

(1)

Multiplying degree-zero localized fractions by a sufficiently large power of the chart denominator reduces the claim to the assumed high-degree isomorphisms.

Relative Proj construction ( $Proj_{X} A$ )

 0  0

For a quasi-coherent graded

O_{X}

-algebra

A

, the relative Proj is obtained by gluing the schemes

Proj A (U)

over affine open subsets

U \subseteq X

. If

A

is of finite type and generated in degree one, a surjection

Sym (E) ↠ A

realizes

Proj_{X} A

as a closed subscheme of the projective bundle

P_{X} (E)

Veronese subring ( $S^{(d)}$ )

 0  0

For a nonnegatively graded ring

S = ⨁_{m \geq 0} S_{m}

and

d \geq 1

, its

d

th Veronese subring is

S^{(d)} = ⨁_{m \geq 0} S_{d m},

(1)

graded by

(S^{(d)})_{m} = S_{d m}

. Passing from

S

S^{(d)}

does not change its projective scheme: the standard affine charts give a canonical isomorphism

Proj S ≅ Proj S^{(d)}

Veronese embedding ( $ν_{d}$ )

 0  0

The degree-

d

Veronese embedding sends a projective point to all monomials of degree

d

in its homogeneous coordinates. For example,

ν_{2} : P^{1} ⟶ P^{2}, [x : y] ⟼ [x^{2} : x y : y^{2}],

(1)

is a closed immersion with image the conic

u w - v^{2} = 0

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Proj construction is likely a reference to "projection construction," which is used in various fields, including computer graphics, engineering, and project management. However, since "Proj construction" could refer to different concepts depending on context, let me outline a few possible interpretations: 1. **Projection Construction in Mathematics/Geometry**: This refers to methods of creating projections of geometric shapes, often to simplify complex visuals or to work within different dimensions.

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Proj construction ( $Proj S$ )

Degree-one generation condition for Proj

Sheaf associated with a graded module ( $M ∣_{D_{+} (f)} = (M_{f})_{0}$ )

Tensor compatibility of graded sheafification on degree-one-generated Proj ( $M \otimes_{O_{X}} N ≅ M \otimes_{S} N$ )

Twisting sheaf on Proj ( $O_{Proj S} (n) = S (n)$ )

Irrelevant ideal of a graded ring ( $S_{+}$ )

Standard affine open of Proj ( $D_{+} (f)$ )

Morphism on Proj induced by a graded ring homomorphism

Invariance of Proj under an eventual graded isomorphism

Relative Proj construction ( $Proj_{X} A$ )

Veronese subring ( $S^{(d)}$ )

Veronese embedding ( $ν_{d}$ )

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